This is probably going to ping some nerves, but I'd say that that's because in the U.S. a good-sized chunk of the white population is actually able to trace their cultural roots back to a specific place. Most black folks don't have that luxury. Related: the movement towards latin@s being treated as an ethnic group as well as a cultural one, which I'd point to several things relating to how Latin America diverged from Spain and Portugal both racially (due to intermingling with Africans and native groups) and to a certain degree culturally.
I'd assume that in, say, Britain, there's also a much firmer divide between racial identity and cultural identity among various immigrant groups. The U.S. doesn't have that as much, partly because even the most well-preserved cultural traditions (i.e. Italians, Irish, Cuban, Korean, &c. immigrant descendents) have homogenized and integrated to a fairly strong degree. Like, I know that I've got a strong chunk of Scots blood in me, but I don't think of myself as Scottish-American despite the fact that a lot of my immigrant ancestors were Scots--that's pre-Revolution days we're talking about, and it's only really hung about in the sense of family heirlooms and stories. The Italians, Irish, &c. immigrant descendents tend to have stronger cultural identities because their ancestors came over more recently and in larger, more interconnected groups, such that they preserved more and haven't had as long to dilute. Meanwhile, to get back to the original point, the average African-American is lucky if they can trace their family back to antebellum days, never mind their ancestors point of origin. Note, also, that actual (that is to say, free-willed) black immigrants from Ethiopia, Jamaica, &c. do (at least in my observation) tend to retain that same sort of cultural identity.
It's definitely affected by duration since the arrival of ancestors, cohesiveness of immigrant groups, and the nature of the process by which they arrived (i.e. immigration vs. slave trade). If I had to hazard a guess, the larger context of the statement would be "I don't identify as 'white' [i.e. homogenous racial group], I'm Italian." Substitute "white" with "European", if that helps. "White" is bland, broad, and not very descriptive; if they're proud of their ancestry, they'd want to display it.
Well, hell, there's hundreds of pages of analysis and explanation in a topic that broad and multi-faceted, but there's a two-cent guess.